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2 Sampling only at a number of locations What is between ?Spatial problemsSampling only at a number of locationsWhat is between ?EstimateQuality of estimationSimulate realizationsGeostatistics (Krige, Matheron)Mining applicationsHydro and Environmental sciences

8 Estimation variance is an index of spatial configurationProblemsEstimation variance is an index of spatial configurationDoes not depend on the local values“Best” for Gaussian distributionSymmetrical (high and low values not distinguished)Variogram estimation difficultSquared differences – skewed distributionDominated by high valuesIndependence of the pairs not fulfilledStrongly influenced by the marginal distribution

17 Indicator variablesInterpretation as probabilityInterpolation of the indicatorsResult  pdf for each locationSimulation restricted to the observed rangeCan copulas be used to overcome some of these problems?

18 How to find such copulas ?Spatial copulasAssumption:Multivariate copula exists for any number of pointsThe bi-variate marginal copulas corresponding to pairs separated by a vector h are translation invariantHow to find such copulas ?

19 Empirical copulasSet of pdf pairs corresponding to points separated by the vector hGeneralization of the variogramEmpirical density using kernel smoothing

33 Copulas and natural processesNatural processes influence high and low values differentlyErosion at high elevationsPollution is spreading not the backgroundWeather relates the high dischargesCopulas of digital elevation models:Spain – eroded old landscapeEcuador – younger but errodedMars – eroded and meteorites

35 Copulas of daily rainfall601 rainfall stations in the Rhein catchment GermanySize = km2Days with important events with good spatial coverage were selected (400 days of the period )Spatial copulas (densities) for different distances were calculated

41 Requirements for a spatial copulaStability of the multivariate marginals: which means that any multivariate marginal copula corresponding to a selected set of points should not depend on the set of other selected points used to define the multivariate copula.Wide range of dependence: a geographically close set of points should have an arbitrarily strong dependence structure, while distant points should be independent.Flexible parametrization: the multivariate copula should have a parametrization such that the dependence structure reflects the geometric position of the corresponding set of points.

54 Parameter estimationNon independent pairs – MLFit the rank correlation function and the asymmetryParametric form of the covariance of the original normalFurther work needed

55 For n+1 points the joint distribution is knownInterpolationFor n+1 points the joint distribution is knownCalculate the conditional for the unobserved pointFull conditional distribution known – thus confidence intervals can be calculatedExample:4 points – corner of a unit squareA: two of them with F(x)=1 two with F(x)=0B: all with F(x)=0.5

62 Validation of the conditional densitiesAre the conditional densities OK ?Cross validationCalculation of the frequencies of non exceedence for the observed valuesComparison with the uniformV is much better then normal or Kriging

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